我从散点图中提取数据进行分析，发现多项式 + 正弦似乎不是最佳模型，因为低阶多项式没有很好地遵循数据的形状，而高阶多项式表现出龙格现象极端数据处的高曲率。我进行了方程搜索以找出可能施加的高频正弦波，一个很好的候选者似乎是极值峰值方程“a * exp(-1.0 * exp(-1.0 * ((xb)/c ))-((xb)/c) + 1.0) + offset" 如下图。
这是这个方程的图形 Python 曲线拟合器，在文件的顶部我加载了我提取的数据，因此您需要用实际数据替换它。该拟合器使用 scipy 的差分进化遗传算法模块来估计非线性拟合器的初始参数值，该拟合器使用拉丁超立方算法来确保对参数空间的彻底搜索，并需要搜索范围。这里的界限取自数据的最大值和最小值。
从此拟合曲线中减去模型预测应该只剩下要建模的正弦分量。我注意到在大约 x = 275 处似乎还有一个窄的、低振幅的峰值。
import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import warnings
##########################################################
# load data section
f = open('/home/zunzun/temp/temp.dat')
textData = f.read()
f.close()
xData = []
yData = []
for line in textData.split('\n'):
if line: # ignore blank lines
spl = line.split()
xData.append(float(spl[0]))
yData.append(float(spl[1]))
xData = numpy.array(xData)
yData = numpy.array(yData)
##########################################################
# model to be fitted
def func(x, a, b, c, offset): # Extreme Valye Peak equation from zunzun.com
return a * numpy.exp(-1.0 * numpy.exp(-1.0 * ((x-b)/c))-((x-b)/c) + 1.0) + offset
##########################################################
# fitting section
# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
warnings.filterwarnings("ignore") # do not print warnings by genetic algorithm
val = func(xData, *parameterTuple)
return numpy.sum((yData - val) ** 2.0)
def generate_Initial_Parameters():
# min and max used for bounds
maxX = max(xData)
minX = min(xData)
maxY = max(yData)
minY = min(yData)
minData = min(minX, minY)
maxData = max(maxX, maxY)
parameterBounds = []
parameterBounds.append([minData, maxData]) # search bounds for a
parameterBounds.append([minData, maxData]) # search bounds for b
parameterBounds.append([minData, maxData]) # search bounds for c
parameterBounds.append([minY, maxY]) # search bounds for offset
# "seed" the numpy random number generator for repeatable results
result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
return result.x
# by default, differential_evolution completes by calling curve_fit() using parameter bounds
geneticParameters = generate_Initial_Parameters()
# now call curve_fit without passing bounds from the genetic algorithm,
# just in case the best fit parameters are aoutside those bounds
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('Fitted parameters:', fittedParameters)
print()
modelPredictions = func(xData, *fittedParameters)
absError = modelPredictions - yData
SE = numpy.square(absError) # squared errors
MSE = numpy.mean(SE) # mean squared errors
RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))
print()
print('RMSE:', RMSE)
print('R-squared:', Rsquared)
print()
##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
axes = f.add_subplot(111)
# first the raw data as a scatter plot
axes.plot(xData, yData, 'D')
# create data for the fitted equation plot
xModel = numpy.linspace(min(xData), max(xData))
yModel = func(xModel, *fittedParameters)
# now the model as a line plot
axes.plot(xModel, yModel)
axes.set_xlabel('X Data') # X axis data label
axes.set_ylabel('Y Data') # Y axis data label
plt.show()
plt.close('all') # clean up after using pyplot
graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)
更新 --------
如果高频正弦分量是恒定的（我不知道），那么只用几个周期对一小部分数据进行建模就足以确定拟合模型正弦波部分的方程和初始参数估计值。在这里，我这样做了，结果如下：
根据以下等式：
amplitude = -1.0362957093184177E+00
center = 3.6632754608370377E+01
width = 5.0813421718648293E+00
Offset = 5.1940843481496088E+00
pi = 3.14159265358979323846 # constant not fitted
y = amplitude * sin(pi * (x - center) / width) + Offset
使用实际数据而不是我的散点图提取数据组合这两个模型应该接近您的需要。